{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/sparse-recovery-via-differential-inclusions","title":"Sparse Recovery via Differential Inclusions","arxiv_id":"1406.7728","date":"2014-06-30","proceeding":null,"authors":["Stanley Osher","Feng Ruan","Jiechao Xiong","Yuan YAO","Wotao Yin"],"abstract":"In this paper, we recover sparse signals from their noisy linear measurements\nby solving nonlinear differential inclusions, which is based on the notion of\ninverse scale space (ISS) developed in applied mathematics. Our goal here is to\nbring this idea to address a challenging problem in statistics, \\emph{i.e.}\nfinding the oracle estimator which is unbiased and sign-consistent using\ndynamics. We call our dynamics \\emph{Bregman ISS} and \\emph{Linearized Bregman\nISS}. A well-known shortcoming of LASSO and any convex regularization\napproaches lies in the bias of estimators. However, we show that under proper\nconditions, there exists a bias-free and sign-consistent point on the solution\npaths of such dynamics, which corresponds to a signal that is the unbiased\nestimate of the true signal and whose entries have the same signs as those of\nthe true signs, \\emph{i.e.} the oracle estimator. Therefore, their solution\npaths are regularization paths better than the LASSO regularization path, since\nthe points on the latter path are biased when sign-consistency is reached. We\nalso show how to efficiently compute their solution paths in both continuous\nand discretized settings: the full solution paths can be exactly computed piece\nby piece, and a discretization leads to \\emph{Linearized Bregman iteration},\nwhich is a simple iterative thresholding rule and easy to parallelize.\nTheoretical guarantees such as sign-consistency and minimax optimal $l_2$-error\nbounds are established in both continuous and discrete settings for specific\npoints on the paths. Early-stopping rules for identifying these points are\ngiven. The key treatment relies on the development of differential inequalities\nfor differential inclusions and their discretizations, which extends the\nprevious results and leads to exponentially fast recovering of sparse signals\nbefore selecting wrong ones.","url_abs":"http://arxiv.org/abs/1406.7728v5","url_pdf":"http://arxiv.org/pdf/1406.7728v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"sparse-recovery-via-differential-inclusions","repo_url":"https://github.com/yuany-pku/split-lbi","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}